Study describes how conformal metrics behave as Q-curvature changes, forming spherical bubbles.
problem Understanding the asymptotic behavior of conformal metrics with null Q-curvature.
method Analyzes the GJMS operator and uses higher order Bol's inequality to describe the metrics' behavior.
result Normalized conformal metrics form exactly one spherical bubble as λ approaches zero.
The study solves a problem in conformal geometry with applications to Q-curvature.
problem Existence of solutions to conformally invariant equations.
method Volume comparison theorems and volume rigidity theorems with respect to Q-curvature.
result Sufficient and necessary conditions for the existence of solutions to conformally invariant equations.
New Bol operators found for supermanifolds with specific dimensions.
problem Finding invariant differential operators for supermanifolds.
method Described analogs of Bol operators for pgl(a+1∣b)-invariant differential operators. result Many new differential operators discovered for (a∣b)=(2∣0),(0∣3) and 1∣1-dimensional superstring. Our aim in this paper is to classify the 3-dimensional connected differentiable global Bol loops, which have a non-solvable group as the group topologically generated by their left translations and to describe their relations to metric space geometries. The classification of global differentiable Bol loops significan…
We equip many non compact non simply connected surfaces with smooth Riemannian metrics whose isoperimetric profile is smooth, a highly non generic property. The computation of the profile is based on a calibration argument, a rearrangement argument, the Bol-Fiala curvature dependent inequality, together with new result…
Using the relations between the theory of differentiable Bol loops and the theory of affine symmetric spaces we classify all connected differentiable Bol loops having an at most 9-dimensional semi-simple Lie group as the group topologically generated by their left translations. We show that all these Bol loops are is…
New Bol operators identified on superstrings.
problem Classifying Bol operators on superstrings.
method Invariant classification of Bol operators on supermanifolds.
result Many new Bol operators discovered.
The paper improves CR Sobolev inequalities and classifies minimizers.
problem Higher-order CR Sobolev inequalities on the CR sphere.
method Improvement through vanishing higher order moments of the volume element.
result New direct proof of minimizers' classification and existence of minimizers in C2k(N). The fundamental ideas of the definition of solvable and semisimple Bol algebras are given and some related theorems
Unified approach to discrete and smooth isoperimetric inequalities of arbitrary order.
problem Finding higher order isoperimetric inequalities for both discrete and smooth curves.
method Unified approach via Fourier analysis of linear operators.
result Unified upper and lower bounds for isoperimetric deficit in smooth curves.
The paper introduces boundary operators for Poincaré-Einstein manifolds and proves higher order trace inequalities.
problem Proving higher order trace inequalities on Poincaré-Einstein manifolds.
method Introducing conformally covariant boundary operators and using them to set up higher order Dirichlet problems.
result Sharp higher order Sobolev trace inequalities on the ball are obtained.
The fundamental ideas of aplicability of Levi-Malcev Theorem for Bol algebras, which plays a basic role in structural theory are outlined
Study initiates homology theory for Bol-Moufang quasigroups.
problem Developing a homology theory for a specific type of quasigroups.
method Using extensions by affine quasigroups, define boundary operations and compute homology groups.
result Computed second homology groups for examples, speculating on relation to functorial homology.
Sharp inequalities in unit ball with constraints on moments.
problem Establishing Sobolev trace inequalities with constraints.
method Constructing smooth test functions for higher order moments.
result Almost optimal Sobolev trace inequalities for 2nd and 4th orders.
Motivated by a recent work of Ache and Chang concerning the sharp Sobolev trace inequality and Lebedev-Milin inequalities of order four on the Euclidean unit ball, we derive such inequalities on the Euclidean unit ball for higher order derivatives. By using, among other things, the scattering theory on hyperbolic space…
Sharp spectral gap estimates for higher-order operators on hyperbolic spaces.
problem Estimating spectral gaps for higher-order operators on Cartan-Hadamard manifolds.
method Symmetrization-free proofs based on general functional inequalities.
result Solves a sharp asymptotic problem from Cheng and Yang and answers a question from Kristály.
We prove that the isoperimetric inequalities in the euclidean and hyperbolic plane hold for all euclidean, respectively hyperbolic, cone-metrics on a disk with singularities of negative curvature. This is a discrete analog of the theorems of Weil and Bol that deal with Riemannian metrics of curvature bounded from above…
Improved algorithms for convex-concave min-max optimization and monotone variational inequalities.
problem Efficiently solving constrained convex-concave min-max problems and monotone variational inequalities.
method Higher-order methods achieving iteration complexities of O(1/T^{ rac{p+1}{2}}) for p-th order derivatives.
result Achieved improved convergence rates for min-max and monotone variational inequalities.
The paper derives new inequalities on manifolds and applies them to convex hypersurfaces.
problem Deriving new inequalities on manifolds and convex hypersurfaces.
method Using Fourier theory and geometric implications of Poincare-type inequalities.
result Sharp Minkowski-type inequalities, including stability and Alexandrov-Fenchel inequalities.
The paper extends inequalities for convex bodies to higher dimensions and various norms.
problem Extending inequalities for convex bodies to higher dimensions and various norms.
method Developed new operators and inequalities for higher-order Lp norms. result Established mth-order Lp isoperimetric inequalities. New Gini indices capture more nuanced income inequality.
problem Measuring joint dispersion across multiple observations.
method Axiomatic approach to define and characterize n-th order Gini deviations.
result Higher-order Gini coefficients reveal more extreme income disparities.
Disproves Fedotov's conjecture on higher-order Shephard inequalities.
problem Fedotov's conjecture on higher-order Shephard inequalities.
method Using Hodge-Riemann relations for simple convex polytopes.
result Fedotov's conjecture is disproved.
We prove a lower bound for the k-th Steklov eigenvalues in terms of an isoperimetric constant called the k-th Cheeger-Steklov constant in three different situations: finite spaces, measurable spaces, and Riemannian manifolds. These lower bounds can be considered as higher order Cheeger type inequalities for the Ste…
The paper introduces new boundary operators and proves higher order CR Sobolev trace inequalities for Siegel domain and complex ball.
problem Establishing higher order CR Sobolev trace inequalities for Siegel domain and complex ball.
method Introducing conformally covariant boundary operators, proving extension theorems, and establishing trace inequalities.
result Generalized CR Sobolev trace inequalities for all γ ∈ (0, n+1) \mathbb{N}.
We prove that a d-web near a point in n-space, where n is greater than 2 and d is greater than 2n-1, is equivalent to an algebraic web, if it has maximal rank or, more generally, if it has (2d - 3n + 1) abelian relations the 1-jets of which are linearly independent. In case n=3, this is a theorem of Bol. The general ca…
New proof of Sobolev inequality with constraints on sphere.
problem Improving Sobolev inequality on sphere with constraints.
method Careful study of extremal problem on sphere.
result Explicit determination of constant in second order moments case.
Sharp bounds for curve isoperimetric deficit derived.
problem Finding sharp bounds for the isoperimetric deficit of curves.
method Fourier analysis applied to derive Wirtinger-type inequalities.
result Sharp lower and upper bounds for the isoperimetric deficit proved.
We prove some isoperimetric type inequalities in warped product manifolds, or more generally, multiply warped product manifolds. We then relate them to inequalities involving the higher order mean-curvature integrals. We also apply our results to obtain sharp eigenvalue estimates and some sharp geometric inequalities i…
A classical result of Aubin states that the constant in Moser-Trudinger-Onofri inequality on S2 can be imporved for furnctions with zero first order moments of the area element. We generalize it to higher order moments case. These new inequalities bear similarity to a sequence of Lebedev-Milin type inequa…
The biharmonic flow and Willmore flow are studied in higher dimensions using geometric evolution equations.
problem Prove global existence of the Willmore flow in higher dimensions.
method Apply Michael-Simon-Sobolev inequality and Gagliardo-Nirenberg inequalities to establish local energy estimates and maximal existence time.
result Global existence of the Willmore flow in higher dimensions is proven.
Solved Cheeger inequalities for simplicial complexes, combining topological and graph theoretic methods.
problem Extend Cheeger inequalities to simplicial complexes and their higher order Laplacians.
method Combining constructions from simplicial topology, signed graphs, Gromov filling radii, and interpolating between 1-Laplacians and 2-Laplacians.
result Developed a general theory for p-Laplacians on simplicial complexes and proved Cheeger-type inequalities.
We consider a family of C1 vector fields satisfying a suitable higher order involutivity condition. We discuss the definition of commutators, the regularity of Sussmann's orbits and the Poincaré inequality.
In this paper, we establish some sharp inequalities between the volume and the integral of the k-th mean curvature for k+1-convex domains in the Euclidean space. The results generalize the classical Alexandrov-Fenchel inequalities for convex domains. Our proof utilizes the method of optimal transportation.
New method proves inequalities for self-shrinkers using perturbation.
problem Proving Łojasiewicz inequalities for self-shrinkers.
method Perturbative analysis of a new auxiliary quantity.
result New method interpolates between higher order and differential geometric approaches.
This paper establishes inequalities on quaternionic hyperbolic spaces and the Cayley hyperbolic plane.
problem Establishing higher order Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities on quaternionic hyperbolic spaces and the Cayley hyperbolic plane.
method Developing factorization theorems and introducing Geller's operators, combining with Helgason-Fourier analysis and kernel estimates.
result Established higher order Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities on quaternionic hyperbolic spaces and the Cayley hyperbolic plane.
We list up to Möbius equivalence all possible degrees and embedding dimensions of real surfaces that are covered by at least two pencils of circles, together with the number of such pencils. In addition, we classify incidences between the contained circles, complex lines and isolated singularities. Such geometric chara…
We bound the higher-order Dehn functions and other filling invariants of certain Carnot groups using approximation techniques. These groups include the higher-dimensional Heisenberg groups, jet groups, and central products of two-step nilpotent groups. Some consequences of this work are a construction of groups with ar…
Sharp inequalities for radial functions on hyperbolic spaces without boundary conditions.
problem Establishing inequalities for radial functions on hyperbolic spaces without zero boundary conditions.
method Novel approach considering both bounded and unbounded domains, focusing on weighted Sobolev and Adams-Trudinger-Moser embeddings.
result Theorems 1.2, 1.3, and 1.4 for weighted Sobolev embedding theorems, and Theorems 1.5 and 1.6 for Adams-Trudinger-Moser type embedding theorems.
We prove sharp blow up rates of solutions of higher order conformally invariant equations in a bounded domain with an isolated singularity, and show the asymptotic radial symmetry of the solutions near the singularity. This is an extension of the celebrated theorem of Caffarelli-Gidas-Spruck for the second order Yamabe…
Improved CR Sobolev inequalities on CR sphere established.
problem Establishing CR Sobolev inequalities on CR sphere.
method Nice commutator identities involving CR intertwining operators.
result Simpler proof of existence and classification of minimizers.
The article proves a Poincaré inequality for hypersurfaces and applies it to rigidity results.
problem Proving rigidity results for hypersurfaces under curvature constraints.
method Using a divergence formula for symmetric endomorphisms, the article deduces a Poincaré type inequality and applies it to higher-order mean curvature of hypersurfaces.
result The article proves several rigidity results for complete r-minimal hypersurfaces.
This paper proves a Liouville type result for a specific higher-order equation on the sphere.
problem Proving that positive, smooth solutions to a particular equation on the sphere are constant under certain conditions.
method Analyzing the GJMS operator and using Liouville type results.
result Positive, smooth solutions to the equation are constant under specified conditions.
The paper provides bounds for high-dimensional U-statistics with novel order-explicit inequalities.
problem Bounding the deviation of high-dimensional U-statistics from their Hájek projections.
method Develops novel order-explicit moment inequalities for higher-order Hoeffding components.
result The maximum deviation of a high-dimensional U-statistic from its Hájek projection is of order Op(φbn−1log2(dn)). We study the Unadjusted Langevin Algorithm (ULA) for sampling from a probability distribution ν=e−f on Rn. We prove a convergence guarantee in Kullback-Leibler (KL) divergence assuming ν satisfies a log-Sobolev inequality and the Hessian of f is bounded. Notably, we do not assume convexity or boun…
This article is an attempt to generalize Riemann's bilinear relations on compact Riemann surface of genus at least 2, which may lead to new structures in the theory of hyperbolic Riemann surfaces. No significant result is obtained, the article serves to bring the readers' attention to the observation made by [Bol-1949]…
Note on failure of Martingale Wasserstein Inequality in higher dimensions.
problem Analyzing failure of Martingale Wasserstein Inequality in higher dimensions.
method Checking failure in dimension d≥2 and proving a stronger inequality in all dimensions.
result A stronger Maximal Martingale Wasserstein Inequality holds in all dimensions.
The paper proves new inequalities on the unit ball in higher dimensions.
problem Establishing new weighted inequalities on the unit ball.
method Limiting approach to prove Carleman and Huber inequalities.
result Sharp weighted Carleman and Huber inequalities on the unit ball.
New algorithm samples superlinearly growing log-gradient distributions.
problem Sampling from distributions with superlinearly growing log-gradient.
method Proposes a novel taming Langevin-based scheme called sTULA.
result Derives non-asymptotic convergence bounds in KL, TV, and W2 distances.